Operator Representation and Logistic Extension of Elementary Cellular Automata

Operator representation and logistic extension of elementary cellular automata M. Ibrahimia, A. Guçl¨ u¨b,c, N. Jahangirovb, M. Yamand, O. Gulseren¨ b,e, S. Jahangirovb,f aCentre de Physique Théorique(CPT), Turing Center for Living Systems, Aix Marseille Université,13009 Marseille, France bUNAM-Institute of Materials Science and Nanotechnology, Bilkent University, Ankara 06800, Turkey cDepartment of Electrical and Electronics Engineering, Bilkent University, 06800, Ankara, Turkey dDepartment of Aeronautical Engineering, University of Turkish Aeronautical Association, 06790, Ankara, Turkey eDepartment of Physics, Bilkent University, Ankara 06800, Turkey fInterdisciplinary Graduate Program in Neuroscience, Bilkent University, 06800, Ankara, Turkey Abstract We redefine the transition function of elementary cellular automata (ECA) in terms of discrete operators. The operator representation provides a clear hint about the way systems behave both at the local and the global scale. We show that mirror and complementary symmetric rules are connected to each other via simple operator transformations. It is possible to decouple the representation into two pairs of operators which are used to construct a periodic table of ECA that maps all unique rules in such a way that rules having similar behavior are clustered together. Finally, the operator representation is used to implement a generalized logistic extension to ECA. Here a single tuning parameter scales the pace with which operators iterate the rules. We show that, as this parameter is tuned, many rules of ECA undergo multiple phase transitions between periodic, locally chaotic, chaotic and complex (Class 4) behavior. Emergence, a semantic gap between behavior and interac- is a simple, yet often efficient approach for predicting the class tions, is a hallmark of dynamical systems. Examples of emer- [8]. Also, further approaches have introduced entropies, mean gent behavior include: exchanging deals between thousands field descriptions or network analyses that help to understand of agents/companies making up the whole stock market; inter- the relation between patterns and their respective rules [9, 10, actions between alternately expressing genes generating juxta- 11, 12, 13]. However, these analyses are either too generic to posed biological forms; series of firings between peculiarly in- hold for, or too specific to apply to larger families of CA. Con- terconnected neurons breeding functional activities in the brain. sequently, the quest for generalizing a method to any dynamical Given the particularities of such systems (multiple levels of lattice system remains challenging and this motivates the need hierarchies, events and processes operating at broadly differ- to approach CA rules in the light of a different perspective. ent space/time scales), complexity science has adopted cellu- Using the ECA set as an example, we suggest a fundamen- lar automata (CA) [1] as much simpler computational models tal approach that redefines the transition function based on a to specifically target the semantic gap between individual and simple intuition gained by visual inspection of the system scale global degrees [2, 3, 4]. These agent based models operate in dynamics. This approach brings the microscopic information fully discrete domains and are known to generate large scale closer to the large scale dynamics and thus helps understand types of behavior only through local interactions (rules) [5, 6]. the properties of a system based on its “first principles”. While aiming to acquire a generic understanding applicable Having distilled the interactions into iterative operations, to all dynamical systems, a main hypothesis has gathered sev- we employ these operations and the symmetries of the system to eral attempts to bridge the spatio-temporal patterns observed rewrite the transition function in a more intuitive notation. Our in CA (phenotype) with their rule space (genotype). The lead- approach provides a framework that links the similarities and arXiv:2010.02073v1 [nlin.CG] 5 Oct 2020 ing apprehension is Wolfram’s classification which postulates differences observed at the phenotype level to an operator based that the asymptotic behavior of a dynamical system lies in one translation of the rule space. Furthermore, this framework en- of these four classes: homogeneous, periodic, aperiodic and ables us to implement a generalized logistic extension to ECA complex behavior [2, 7]. He introduced elementary cellular [14] where a single parameter scales the pace with which op- automata (ECA) as a paradigmatic simple set of rules which erators iterate the system. As a result, the binary state space of comprise all these types. ECA is expanded into a Cantor set and in turn we get a chance to Several studies have attempted to understand how distinct observe transitions between classes [5]. In particular, we reveal groups of rules act to generate similar types of asymptotic be- several complex (Class 4) instances that are not reachable in havior, eventually making up the four classes. Langton’s method the standard ECA. More interestingly, the behavioral difference of labelling a parameter out of a unit interval to a certain rule between some rules sharing similar genetic code is diminished upon the logistic extension. ECA are time dependent one dimensional infinite strings Email address: [email protected] (S. Jahangirov) Preprint submitted to Elsevier October 6, 2020 t t 1 of sites S = fsngjn=−∞ of a binary state space si 2 f0; 1g. In ECA a Rule defines how the value of a certain site is iterated t+1 t si = fS si based on its current value and the values of its t t t nearest neighbors, through a transition function fS (si−1; si; si+1). Given the binary state space, there are eight possible configurations of a three site neighborhood, resulting in 28 = 256 possible mappings, i.e rules. Mapping of Rule 30 is shown as an example in Fig. 1(a). The name “30” of this rule comes from the binary to decimal transformation of the string 00011110 obtained from the particular mapping of the eight configurations listed in the order shown in Fig. 1(a). ECA possesses two important symmetries: complementary and mirror. Here complementary means flipping the state in every site of the array. Hence, if Rule A and Rule B are complementary (mirror) symmetric, then running Rule A with a certain initial string will give the complementary (mirror) image of running Rule B with the complementary (mirror) version of that string. When these symmetries are taken into account, the number of unique rules reduces to 88 (and not 64 since mirror and/or complementary symmetries of certain rules are equivalent to themselves). Simple observations on ECA runs reflect visual structures Figure 1: (a) Representation of the Rule 30 in terms of operators. (b) Trans- of uniform, stable, oscillatory or irregular patterns. These struc- formations needed to switch between mirror and complementary symmetries of a rule. (c) Switching between the Rule 30 and its symmetries using operator tures are prone to a mixture of three types of fundamental iter- representation. t t+1 ations [si ! si ], namely: decay [0j1 ! 0], stability [0(1) ! 0(1)], and growth [0j1 ! 1]. Note that this approach becomes evident in a numerical representation of the state space, and it is under mirror transformation. Omitting one of these pairs erases the key in translating the rules into discrete operators. We first a whole column of repetitions. Continuing in this fashion one use the mirror symmetry of these systems to regroup the eight can reach at a 10×10 table that has all 88 unique Rules with possible configurations into symmetric and asymmetric sets, as only 12 repetitions. While constructing this table, one needs to seen in Fig. 1(a). Then, within each set, we group complemen- decide which repeating columns to erase and how to arrange the tary configurations together, leading to four groups (denoted by rows and the columns that are left at the end. The table that we Roman numerals in four different colors). Each group has cen- have constructed, after evaluating numerous options based on tral cells with values 0 and 1 that are mapped in four different mathematical and aesthetic criteria, is presented in Fig. 2. The ways: [0 ! 0, 1 ! 0], [0 ! 0, 1 ! 1], [0 ! 1, 1 ! 0], and 12 repetitions that appear at the corners of the table are removed [0 ! 1, 1 ! 1]. We call these double mappings operators, and for clarity. Note that, every adjacent row and column share at conveniently name them as Decay (D), Stability (S), Oscilla- least one common operator which means that every adjacent tion (O), and Growth (G), respectively. Note that the oscillation rule on the Table share at least three common operators. operator is a compound of decay and growth iterations. Four The Periodic Table presented in Fig. 2 offers a systematic groups and four possible operators cover all 256 rules (44 = 28). “bird’s eye” view of all 88 unique rules of ECA. Rules domi- The operator representation of Rule 30 becomes DSOG. Sym- nated by similar simple patterns (homogeneous, vertical lines, metric counterparts of rules are easily constructed in operator diagonal lines, horizontal stripes) tend to appear together. The representation. As shown in Fig. 1(b), to get the mirror symme- rules that show rich behavior populate the “fertile crescent” try of a rule, one needs to switch the operators in the group III along the diagonal where simple rules with contradicting pat- and group IV. To get the complementary of a rule, one needs to terns are expected to overlap. Among these rich rules, the ones replace all D operations (if any) with G and vice versa. Sym- that have common features are also brought together. Rule pairs metries of the Rule 30 (DSOG) found by these transformations 18, 146 and 122, 126 are striking examples of this. Despite the are presented as an example in Fig.

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